When all coefficients of
and
are real (implying that
is the transfer function of
a real filter), it will
always happen that the complex one-pole filters will occur in
complex conjugate pairs. Let
denote any one-pole
section in the PFE of Eq.(6.7). Then if
is complex and
describes a real filter, we will also find
somewhere among
the terms in the one-pole expansion. These two terms can be paired to
form a real second-order section as follows:
Expressing the pole
in polar form as
,
and the residue as
,
the last expression above can be rewritten as
The use of polar-form coefficients is discussed further in the section on two-pole filters (§B.1.3).
Expanding a transfer function into a sum of second-order terms with
real coefficients gives us the filter coefficients for a parallel bank
of real second-order filter sections. (Of course, each real pole can
be implemented in its own real one-pole section in parallel with the
other sections.) In view of the foregoing, we may conclude that every
real filter with
can be implemented as a parallel bank
of biquads.7.6 However, the full generality of a biquad
section (two poles and two zeros) is not needed because the PFE
requires only one zero per second-order term.
To see why we must stipulate
in Eq.(6.7), consider the sum of two
first-order terms by direct calculation:
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(7.9) |